THE 9th INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING May 7-9, 2015

Bucharest, Romania

978-1-4799-7514-3/15/$31.00 ©2015 IEEE

On-line Power Systems Voltage Stability Monitoring using Artificial Neural Networks

Constantin BULAC1, Ion TRIŞTIU1, Alexandru MANDIŞ1, Lucian TOMA1

1University “Politehnica” of Bucharest, Department of Electrical Power Systems cbulac@yahoo.com, ion_tristiu@yahoo.com, alexandrumandis@yahoo.com, lucian.toma@upb.ro

Abstract – A method for on-line voltage stability monitoring of a power system based on Multilayer Perceptron (MLP) neural network is proposed in this paper. Considering that the power system is operating under quasistatic conditions, by using power flow model and singular value decomposition of the reduced Jacobian matrix, a suitable index to quantify the proximity of power system voltage instability is defined. Then, a neuronal network is trained to learn the correlation between the key factors of the voltage stability phenomena and this index. Once trained, the neural network provides the above mentioned voltage stability index as output for a predefined set of input variables that are known as directly influencing the stability conditions of the power system. Since the input variables for the neural network may be obtained from the steady state estimator, the proposed method can be implemented as a function of the Energy Management System (EMS) for on-line voltage stability monitoring. Tests are carried out using the IEEE 30-bus system, where different operating scenarios are considered. Keywords: neural networks, voltage stability index

I. INTRODUCTION

Voltage stability, defined as the ability of a power system to maintain a controllable voltage profile, become a relevant concern during planning and real time operation of the power systems. Even though the phenomenon is recognized as a dynamic one, under certain assumptions, a static system model can be used for voltage stability analysis and monitoring. Static methods investigate the mechanism causing voltage instability and collapse, and are generally based on the load flow equations [1][2][3][4]. Dynamic approaches, based on the observation that the instability and collapse phenomena depend on the dynamics of the loads and voltage control devices, employ generally step-by-step techniques [2][3][4]. An important topic in the area of voltage stability is the voltage security monitoring of power systems. This function is a task that must be performed at regular intervals in the power system control center and can be very time consuming and memory intensive if a comprehensive approach based on the dynamic model is adopted. Consequently, a great deal of effort has been devoted to the development of practical tools for analyzing and monitoring voltage stability of power systems using static approaches. More recently, artificial neural networks (ANNs) have been used to obtain very fast solutions of problems related to power system operating and control, in general, and voltage stability, in particular [5][6]. The main feature of an ANN is the ability to achieve complicated input-output mappings through a learning process, without explicit programming. In

this paper, the effectiveness of using MLP neural network for on-line voltage stability monitoring is explored.

In order to optimize the architecture of the neuronal network and minimize the computation effort, only those state variables with major impact on the power system voltage stability are selected as inputs. The output is the target index, that is the value of the VSI (Voltage Stability Index), defined based on the smallest singular value of the reduced Jacobian matrix [1][2][8].

The input quantities of the neural network may be obtained from the steady state estimator, thereby the proposed method can be implemented as a function of EMS (Energy Management System) for on-line voltage stability monitoring.

The effectiveness of the proposed neural network-based approach has been tested on the IEEE-30 bus test system.

II. STEADY STATE VOLTAGE STABILITY MODEL AND INDEX

The singular value decomposition, one of the methods recommended by the IEEE [1], was employed to identify the key features concerning the voltage stability and to define the voltage stability index (VSI).

Assuming that the power system operates under quasistatic conditions, its state is completely governed by the following set of the nonlinear power flow equations modified to integrate the static characteristics of the loads (voltage dependence of the nodal powers) [2]:

( )

( )

cos sin1

sin cos1

0

0

nU U U G Bi i k ik ik ik iki k

nQ U U U G Bi i k ik ik ik iki k

P θ θ

θ θ

⎡ ⎤⎣ ⎦

⎡ ⎤⎣ ⎦

+ =∑=

− =∑=

−

− (1)

where: Pi and Qi are the active and reactive powers at bus i; Ui and iθ are the voltage magnitude and phase angle at

bus i; ik i kθ θ θ= − .

After linearization of the nonlinear equations (1) around of the actual operating point, the following linear representation of the power system in steady state is obtained:

P PU

Q QU

θ

θ=⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Δθ Δθ ΔPJΔU ΔU ΔQ

J J

J J (2)

where: J is the load flow Jacobian matrix;

ΔP andΔQ = incremental changes in the bus real and reactive powers;

Δθ andΔU = incremental changes in the bus voltages phase angle and magnitude.

The power system voltage stability is affected by both active power P and reactive power Q. However, at each operating point we may keep P constant and evaluate the voltage stability by considering the incremental relationship between Q and U. Based on above considerations we can substitute for 0=ΔP in (2) which give:

R=ΔQ J ΔU (3) where:

1R QU Q P PUθ θ

−= −J J J J J (4) is the reduced Jacobian matrix. If the singular value decomposition [7] [8] is applied to JR, we have:

1

PQnT T

i i ii

σ=

= = ∑J LσR L R (5)

where PQn is the size of RJ (number of PQ buses in the power system), L and R are orthonormal matrices, the singular vectors Li and Ri are the columns of the matrices L and R, and σ is a diagonal matrix of the positive singular values iσ , such that

Qnσσσ ≥≥ ...21 .

If RJ is nonsingular, the effect of a small change in the reactive power injections ΔQ on the ΔU vector can be written as

1 1

1

PQn

T

i i ii

σ− −

=

= = ∑ΔU J ΔQ R L ΔQ (6)

In the vicinity of a voltage collapse point, when a singular value is almost zero, the system U-Q response is entirely determined by the minimum singular value

PQnσ and its

singular vectors PQnL and

PQnR , respectively. Under these

conditions 1 1

PQ PQ PQ

Tn n nσ− −= ≅ΔU J ΔQ R L ΔQ (7)

If PQn=ΔQ L then, from 1

PQ PQ

T

n n =L L and (7) we have 1

PQ PQn nσ −=ΔU R . Therefore the following interpretations can be made [3]:

• the smallest singular value ( )minPQ

n Rσ σ= J , which is a

way of identifying the singularity of the Jacobin matrix, indicates the voltage stability reserve as distance between the studied operation point and the voltage stability limit, where the matrix RJ become singular;

• the right singular vector PQnR corresponding to

Qnσ

indicates the voltage sensitivities (the weak buses);

• the left singular vector PQnL corresponding to

Qnσ

indicates the most sensitive direction for change of reactive power injections.

Based on the above observations concerning the right and the left singular vectors, one can be identified the critical elements of the power system. Thus, if in (6)

PQn=ΔQ L then,

taking into account that 1PQ PQ

T

n n =L L and 0PQ

T

i n =L L for

PQi n≠ , we have

1

PQ PQn nσ −=ΔU R (8)

In addition, if in (2) 0=ΔP , we have

1 1 1

Q QP PU P PU n nθ θ σ− − −Δ = − Δ = −θ J J U J J R (9)

With the phase angle and voltage magnitude variations known, the branches and generators participation factors are calculated as in the modal analysis method [9].

Although the lowest singular value σmin (JR) represents a global index for the voltage instability occurrence, it has the disadvantage of not giving a precise information related to how close is the critical frontier located. In order to have a global information on the location of the equilibrium point with respect to a critical surface, the minimum singular value, σ0min(JR ), for the no-load operating conditions (considered as the most stable [10]), is also calculated. The voltage stability index (VSI) is thereby defined as:

)()(

min0

min

R

RVSIJJ

σσ= (10)

A value of VSI close to zero shows that the power system operating point is located in the vicinity of the critical frontier whereas a value close to 1 shows that the system is far from the critical frontier.

III. NEURAL NETWORK ARCHITECTURE

The Multilayer Perceptron, one of the most popular and successful neural network architectures, consists of a number of identical elements organized in layers, where those on one layer are connected with those on the next layer so that the outputs of one layer are feed-forward as inputs to the next layer (Fig. 1).

Input layerHidden

layer

Output layer

Fig. 1. Structure of a MLP with a hidden layer and a single output

In this work, a MLP-type neural network is proposed for implementation in the power systems control center in order to provide synthetic information regarding the position of the current operating point with respect to a critical frontier, without performing a detailed analysis based on the singular value decomposition of JR, which requires increased computational effort.

The neural network is trained to estimate the value of the

VSI index using information provided by the static state estimator. The architecture of the MLP is shown in Figure 1 and consists of:

• The input layer formed by: 1. Total active and reactive demand (Pd and Qd). 2. Total active and reactive generation (Pg and Qg). 3. Total active and reactive losses (Plosses and Qlosses). 4. Total reactive power generation reserves (Qgr). 5. The number of generators sitting at Qg limit (ngQl). 6. The number of buses with voltage below 1 p.u. (nblv). 7. The number of branches with loading over 75% (nb75). 8. The number of branches out of service (nbo=0 or 1). 9. Voltage magnitude and active and reactive power

demand at the most critical buses (Ui, Pdi, Qdi). 10. Loading of the most critical branches (Sbi). 11. Loading/Reserves of the most critical generators (Qgi).

• The output layer consisting of one neuron that provide the estimated value of VSI.

• A certain number of neurons on the hidden layer which depends on the neuronal network behavior during the training – validation – testing process, in terms of the convergence rate, errors, etc.

In order to classify the actual state of the power system with respect to the surface of critical states, the range [0, 1] of variation of VSI is divided into three subintervals that represent the following classes:

• class C1 characterized by ]( 1,1α∈VSI , corresponding to a stable operation state;

• class C2 characterized by ]( 12 ,αα∈VSI , corresponding to an alert operation state;

• class C3 characterized by [ ]2,0 α∈VSI , corresponding to a critical operation state.

The coefficients α1 and α2 are determined in terms of the power system characteristics based on detailed off-line analyses.

Under the above-mentioned aspects, the method proposed for on-line voltage stability assessment consists in estimating the global VSI by means of a neural network, based on the information achieved from the state estimator. This index can be displayed in the form of a position inside a predefined state range, as shown in Figure 2.

Neu

ral n

etw

ork

The

inpu

t var

iabl

es

ClassificationVSI

α α1 2

Fig. 2. The architecture of the on-line voltage stability monitoring function

using a MLP-type neural network.

IV. SIMULATION RESULTS

The effectiveness of the proposed approach was tested on IEEE 30-bus test system. This system has 6 generators and 24 loads, and 41 branches, respectively (Fig. 3). The base case

load-generation pattern and parameters of the transmissions line/transformers are those provided in [11].

8

752

G1

G5

G6

G2

G3

G4

1 3 4 6

2021

10

2212

9

1116

13

14

15

17

18

23 24

30

29

27

Vulnerable Area

2625

28

19

Fig. 3. IEEE 30 buses test system

The database necessary for training, validation and testing the neural network was generated off-line using the algorithmic software EPSA (Electric Power System Analysis) developed by the authors under MATLAB as an educational and research tool for power system stability analysis.

In the first step, in order to identify the vulnerable network elements and to define the set of input variables, a detailed analysis of the power system behavior under various loading conditions between no-load and maximum was performed. For this purpose, the singular value decomposition method, presented in section II, was implemented in the VoltStab function of the EPSA tool.

Figure 4 shows the variation of the VSI against the loading with respect to the base case, under the N conditions. It can be seen that when the size of the JR matrix changes, following the transition of some PU buses into PQ buses, large steep falls in the VSI value are observed. These falls are also displayed on the voltage stability monitor which warns the operator that critical changes occurred into the power system which force some generators operate at the reactive power limit.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Load level

Vol

tage

Sta

bilit

y In

dex

(VSI

)

Fig. 4. Plot of the VSI versus the load variation.

With the bus sensitivities and the branch and generator participation factors obtained in the first step, the most vulnerable elements were identified:

• buses: 30, 29, 26, 25 and 27 (vulnerable area in Fig. 3). • branches: 27-28, 9-10, 24-25, 27-29, and 27-30. • generators: G3, G5 and G4. Therefore, a number of 34 input quantities for the neural

network were identified, i.e.: 11 global attributes (Pd, Qd, Pg, Qg, ΔP, ΔQ, Qgr, ngQl, nblv, nb75, nbo) and 24 case attributes (5×(Ui, Pdi, Qdi) + 4×Sbi+3×Qgi) .

Then, once the input-output configuration of the neural network is defined, the database was generated. Assuming load variations and/or branch contingency then 7000 sets of data, corresponding to different operating states, were generated. For each operating state, the VoltStab function of the EPSA tool has determined the VSI values based on the singular value decomposition, which represents outputs of the neural network. The architecture of the neural network was achieved with the Neural Network Fitting Tool from Matlab, which helps creating and training a neural network, then evaluating its performances using mean square error and regression analysis.

For the voltage stability analysis, 6000 examples were randomly chosen, out of the 7000 examples initially generated, which were then split such that: 70% were used for training, 15% for validation, and 15% for testing. The Levenberg-Marquardt algorithm was employed for training, considering various numbers of neurons on the hidden layer. In this way, is was identified that the optimum number of neurons on the hidden layer is 15. The neural network performances, evaluated using the mean square error, are presented in Figure 5, and the regression analysis shows a very good input-output correlation, the correlation factor being R=0.99994.

The neural network was tested on the remaining 1000 examples, which proved again a very good input-output correlation, the correlation factor being R=0.99993.

0 100 200 300 400 500 600 700

10-6

10-4

10-2

100Best Validation Performance is 6.5273e-07 at epoch 780

Mea

n Sq

uare

d E

rror

(m

se)

786 Epochs

TrainValidationTestBest

Fig. 5. Neural network performance during the training, validation and

testing process.

V. CONCLUSIONS

The work presented in this paper aimed to analyze the opportunity of employing a MLP-type neural network for on-line power system voltage stability monitoring. The proposed

method assumes performing two steps. In the first step, using an algorithmic procedure based on the singular value decomposition method, a detailed analysis of the power system operation under various conditions with respect to the base case is performed. The results are used to identify the U-Q characteristics of the power system based on which the input parameters of the neural network are defined. In the second step, the neuronal network is generated, trained, validated and tested using the Neural Network Fitting Tool from Matlab to estimate the value of the voltage stability index VSI, based on the smallest singular value of the reduced Jacobian JR. The simulations have shown that one hidden layer only is optimal for fitting the inputs and the output (the VSI) of the neural network. The efficiency of the proposed method was determined on the IEEE 30-bus test system. A software called EPSA, developed by the authors under Matlab, was employed to generate the simulation database. In real control centers, such database can be obtained from the state estimator, thereby the proposed method can be implemented as additional function of the Energy Management System (EMS).

ACKNOWLEDGMENT

The work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132397.

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